Stage de Pratique Accompagnée, M2 Agrégation
Rapport de Stage
Marguerite Flammarion & Elio Joseph 20 décembre 2016
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Stage de Pratique Accompagnée
CRCG
Nous remercions en premier lieu l’Université de Paris Saclay et le département de mathématiques d’Orsay qui, grâce à la formation qu’ils nous proposent, ont donné lieu à ce stage de pratique accompagnée qui se révéla, comme nous le verrons par la suite, enrichissant humainement et professionnellement. Nous sommes reconnaissants à Madame Françoise Issard-Roch qui nous af fecta fecta au lycée lycée René Cassin d’Arpajon d’Arpajon et qui géra géra nos retar retardd avec avec le sourire. sourire. Nous remercions Arpajon, ville agréable tant par son charme que par son calme. Comment ne pas apprécier la qualité environnementale de la ville d’Arpa jon ? Les espaces espaces verts, les pelouses, pelouses, les parterr parterres es fleuris, soigneusement entretenus entretenus par les jardiniers de la ville, incitent à la promenade. Un effort constant est fourni par la commune pour embellir et préserver le cadre de vie. Parmi les récentes réalisations, l’ouverture du parc de la Prairie - espace de détente et de loisirs - permet la poursuite des cheminements et liaisons douces, dans le cadre de la promenade de l’Orge. Nous ne pouvons que remercier madame Marie-Christine Lévi qui nous enseigna l’enseignement et nous apprit à apprendre. Nous ne sommes que ses élèves mais, grâce à elle, nous nous sommes élevés. Toute notre gratitude va à Madame Florence Lefebvre , femme agréable tant par son autorité que par sa bonne humeur. Pédagogue née, nous la remercions pour ses conseils avisés, son partage généreux de sa grande expérience et sa bienveillante supervision. Nous rendons grâce à tous nos professeurs passés, présents et futurs pour nous avoir insufflé la passion de la mathématique et de son enseignement et présenté la muse de la divine inspiration. On le sait maintenant : ils sont toujours présents. Mais nous ne pouvons cesser de nous répandre en remerciements tant leur générosité gratuite et leur constante amitié est hors du commun des mortels. Ils ont relu nos sujets, nos corrections, ont partagé leurs idées, leur expérience, leurs échecs, leur réussite. Nous les bénissons par ordr ordree alph alphab abétiqu étiquee : Julien Julien Brice, Brice, Sandrine Sandrine Gauthier Gauthier,, Florian Florian Granger Granger,, Laurène rène La Latieuletieule- -Cogné, -Cogné, Coralie Coralie Lemonnier. Ils mériteraient mériteraient un par paragr agraphe aphe à eux tous seuls : Gédéon Gédéon Chevallier, le paladin paladin au grand coeur coeur présent aux moments difficiles ; Cyril Falcon, la tête chercheuse qui nous cultiva de sa science et de ses dimensions. Nous remer remercions le lycée lycée René René Cassin Cassin qui nous accepta accepta en stage ; monsieur monsieur le proviseur Bertrand Bertrand Villain Vil lain pour sa confianc confiancee ; mesdames Marie-Fra Marie-Françoise nçoise Gire et Elisabeth Elisabeth Pezon-Sigal pour leur gentil lesse et leur patience patience ; madame Aurélie Aurélie Maupas qui nous permit permit de nous sustenter sustenter aux jours de disette disette ; ainsi que tout le personnel ersonnel
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Stage de Pratique Accompagnée
CRCG
évidemment les professeurs de mathématiques du lycée René Cassin. Nos pensées vont particulièrement à monsieur Jean-François Alliot, madame Séverine Perrache, monsieur Cyril Vargas et madame Sandrine Bozec, qui nous enrichirent notre expérience en nous permettant d’assister à un de leurs cours. Nous exprimons notre sympathie à mademoiselle Marie Escot et monsieur Emmanuel Magnin, dit "Manu", qui nous supportèrent sans relâche et égayèrent nos journées de leurs sourires. Enfin, ces remerciements seraient vains s’ils ne mettaient une fois de plus à l’honneur celle qui fut la véritable actrice de cette aventure. Parfois discrète mais toujours présente par sa poigne de fer et son rire joyeux, elle fut la meilleure des tutrices et restera à jamais dans nos coeurs :
Florence Lefebvre.
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Table des matières matières
Table des matières 1 Planning . . . . . . . . . . . . . . . . . . . . 2 Phase d’observation . . . . . . . . . . . . . . 2.1 Le choix des axes d’observation . . . 2.2 Axe : la discipline . . . . . . . . . . . 2.3 Axe : les diff différents types pes d’activités . 3 Phase de pratique accompagnée . . . . . . . 3.1 Un cours de Elio . . . . . . . . . . . 3.2 Un cours de Marguerite . . . . . . . 3.3 Enseignement . . . . . . . . . . . . . 3.3.1 Notre cours . . . . . . . . . 3.3.2 Le Devoir Maison . . . . . . 3.3.3 Les Travaux Dirigés . . . . 3.3.4 L’interrogation . . . . . . . 3.3.5 Le devoir sur table . . . . . 4 Ce que nous avons appris . . . . . . . . . . . 4.1 En phase d’observation . . . . . . . . 4.2 En phase de pratique accompagnée . 4.2.1 Remarques d’ordre général . 4.2.2 De l’auto-inspection . . . .
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Stage de Pratique Accompagnée
CRCG
Instruisez-le d’exemple et rendez-le parfait, Expliquant à ses yeux vos leçons par l’effet. Corneille, Le Cid, Acte I, Scène 3.
Introduction Laissez-moi Laissez-moi vous vous conter conter une histoire. histoire. . . Autrefois, la riche tribu Ayayonga vivait dans un riche pays couvert de champs fertiles et de prés prés verdoyants. verdoyants. Les Les Ayayonga s’étaient rendus célèbre célèbress non pas pas par le bétail gras qu’ils élevaient, ni par les nombreuses variétés de blé qu’ils cultivaient, mais par le sel gemme qu’ils trouvaient en abondance dans leurs terres. Casser des blocs de sel n’était pas difficile. C’était plus facile, en tout cas, que de cultiver le blé et élever le bétail bétail ! Avec le sel, on pouvait ouvait tout acheter acheter : le blé et le bétail, bétail, de magni fiques tissus, des bijoux et des armes. Ainsi, les Ayayonga abandonnaient abandonnaient peu à peu le travail, négligeant leurs terres et leurs bêtes, entièrement occupés à l’extraction du sel qui était la source de leurs richesses. Le puissant dieu Mguri-mgori considérait avec mécontentement l’attitude des Ayayonga. Il décida, en fin de compte, de les punir pour leur paresse. Mais auparavant, il voulait leur accorder une chance. Ainsi, un vieillard nommé Bakomba s’adressa au peuple ayayonga : "Ecoutez-moi, hommes ayayonga ayayonga ! Je suis assez vieux, vieux, hommes, hommes, pour me permettr ermettree de vous parler parler comme comme le puissant Mguri-mgori me l’a ordonné. Le puissant dieu est en colère contre vous, car vous négligez votre bétail et ne cultivez plus votre terre. Vous ne faites qu’extraire le sel et amasser amasser les richesses, richesses, oubliant oubliant vos dieux, y compris ompris le grand Mguri-mgo Mguri-mgori ri !" Bakomba n’eut pas le temps de finir. Les Ayayonga se jetèrent sur lui et l’accusèrent de convoitise et d’ambition. Ils décidèrent de s’en débarrasser. Ils attachèrent le pauvre vieillard à un arbre et le lapidèrent avec des blocs de sel. Avant de mourir, Bakomba s’écria : "Le puissant Mguri-mgori vous regarde et son oeil affligé détruira votre votre arro arrogante gante richesse richesse !" A peine eut-il pronon prononccé ces paroles aroles que le ciel se couvrit couvrit de lourds nuages noirs qui commencèrent à déverser des trombes d’eau sur la terre. Dans son affliction, Mguri-mgori inonda de ses larmes le pays des Ayayonga, faisant fondre fondre leur richesse, le sel. Lo Lorsque rsque le soleil brilla à nouveau nouveau dans le ciel, il ne restait restait rien de la tribu ayayonga, de ses champs et pâturages, de ses mines de sel. A leur place s’étendaient à perte de vue des marais salés, dans lesquels aucune végétation ne pousse, aucun poisson, aucun crocodile ne vit. On appelle ces marais Topé-tope, "Larmes salées".
Ce conte nous enseigne que la paresse et la négligence sont rarement récompensées. C’est donc avec les conseils du sage Bakomba en tête que nous avons effectué notre stage de pratique encadrée, au cours de notre seconde année de Master de formation formation des (futurs) (futurs) professeurs agrégés – Parcours Parcours Mathématiques. Mathématiques.
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Stage de Pratique Accompagnée
CRCG
Sujet du Stage age Stag Stagee de pratique acc accomp ompagné gnée Lieu du Sta Stage Lycée René Cassin, à Arpajo ajon (91) Dates Dates du stag stagee Les lundis lundis du 10 10 octobr octobree au 12 décem décembre bre 2016 Duré Duréee du stag stagee 32 heur heures es d’ob d’obse serv rvat atio ionn et de prat pratiq ique ue Classes obs observées De la secon econdde à la terminal nale Clas Classe se ense enseig igné néee Termi ermina nale le Scie Scienntifiq tifique ue numér uméroo 4 Tutrice du stage Madame Florence Lefebvre Parlons un peu du lycée qui nous a accueilli pour notre stage. Le lycée René Cassin n’a pas de biographie sur le site de l’Académie de Versailles. En revanche, nous pouvons donner une idée de son niveau : 1526 sur les 2277 lycées en 2016 (ce classement évoluera possiblement en 2017), mais aussi 25 sur les 42 lycées du département la même année. Enfin, l’Express lui accorde la note de 14 1 4,4/20 en méthodologie, thodologie, soit 18 1 8/25 en écriture fractionnaire.
Figure
vide
1 – La salle des profs,
2 – Le lycée René Cassin Figure
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Stage de Pratique Accompagnée
Figure 3
– R.C.
CRCG
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Stage de Pratique Accompagnée
CRCG
le reste de sa vie. Il prit alors une décision héroïque : celle de rendre la guerre horsla-loi. Dans la seconde partie de sa vie, René Cassin eut une influence déterminante au sein de la France libre, puis en droit international public. Il passa l’arme à droite (il était gaucher) le 20 février 1976 à Paris. Finalement, ce que René Cassin laissa à la postérité fut son fils : Marc. En phase d’observation, nous avons observé plusieurs classes : une terminale scientifique spécialité mathématiques, une première scientifique, une seconde générale, une terminale scientifique spécialité ISN et enfin une terminale scientifique normale. Nous n’avons pas observer de première STMG suite à un imprévu de dernière minute. En phase de pratique accompagnée, nous avons pris en charge la classe numéro quatre de terminale scientifique pour leur enseigner les nombres complexes. Nous avons observé un nombre d’heure certain, nous avons enseigné environ seize heures. Le contact avec notre tutrice fut aisé. Nous l’avons appelée par voie cellulaire afin de convenir d’un rendez-vous, qui fut fixé au premier lundi de notre stage. Nous pénétrâmes dans la salle des professeurs et là, nous la vîmes. Nous balbutiâmes un "euh... bonjour... nous cherchons madame Lefebvre ". ". "C’est MOI," nous réponditelle d’une voix profonde. Nous fîmes connaissance. connaissance. Tout Tout de suite, elle nous plut. "Je ne considère pas être une prof modèle affirmait-elle ; pourtant, à nos yeux, elle modèle " nous affirmait-elle l’était. Ainsi, nous passâmes deux mois incroyables dans cette aventure obscurément lumineuse.
Florence Flore, anse Le fait vraiment
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Stage de Pratique Accompagnée ; ; . t s e n o t r n o i i s e a i d f b a i n D s m i e o b l c s e s m i l n l e o i l c e e n l a D u o c e i e x d r e n fl t o é a i t t r l a l v a u s r é r e s e b l o ’ e L fi i r é v e c n e i r é p x e ’ l
4 4 S S T T s s b b o o é S p s e r S è T 1 s s b b o o é p s N S S I T é n s p o i t S a T v r s e s b b o o e c i r ) t u h t 2 ( c e 4 v a S n T e s i t b e r o t n e 1 2
s e x e l p m o c s e r b m o n s e l s u o t r u s e m m a r g o r p e l t u o t e d e r t i p a h c e l t u o t e d s r u o c e l t u o t e d n o i t a r a p
h 1 h g r g 1 a r i o M a l E D M D T D T T e c i r t u P é t p s c A e S e v d n T a 2 n s s e b i b o t o e r t n e n e i t e r h 1 o t n i e g l r a E t s e M r g r u s r o a u c M o c s r u o c h o g r 1 i l a o E i l s M s E r r u u s o o r c c u o S E c N I E O R R I T T
CRCG
e e s o u i e l d n i r E 2 o l s D b g T O n fi n e i t e r t n e
n e i t p e a r j t n e
e l h b 1 t a s e g r r l a s u a n M r n i A o D v T e d o i l e E l s b r a u t s e o l c r u s a n O r n N i A o O v G e I D R T 6 7 8
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Stage de Pratique Accompagnée
2
CRCG
Phas Ph ase e d’ d’ob obse serv rvat atio ion n L’expérience s’achète par le malheur. Honoré de Balzac
Nous avons observé différentes classes, allant de la seconde générale et technologique nologique à la terminale terminale scientifique. scientifique. Plus précisément, nous avons observé quatre heures 1 de cours en une classe de terminale terminale scientifique, scientifique, cette classe-même classe-même dans laquelle laquelle nous allions enseigner par la suite. Nous avons également observé trois heures de cours d’option de spécialité mathématique en classe terminale scientifique. Ces sept heures d’observation se firent pendant les cours de notre tutrice, madame Florence Lefebvre. Mais ce n’est pas tout, et c’est loin d’être fini, nous avons également observé une heure de cours d’option de spécialité I.S.N., en classe de terminale scientifique, avec l’aimable autorisation du talentueux monsieur Alliot Jean-François. Ultérieurement, nous avons observé une heure de cours en classe de première scientifique, en compagnie de nos camarades de classe sous la tutelle de madame Perrache Séverine. Subséquemment, nous avons observé une heure de cours d’A.P. (lire "Aide Personnali Personnalisée"), sée"), en classe de seconde générale et technologiq technologique, ue, plus exactement exactement un tiers de celle-ci 2 , grâce au chaleureux accueil de l’aimable monsieur Vargas Cyril. Enfin, nous avons observé une heure de cours en classe de seconde générale et technologique, cette fois-ci en classe entière, grâce à la touchante hospitalité de madame Bozec Sandrine. Sandrine. Il était initialement prévu que nous observassions une heure de cours en classe de première première S.T.M.G., S.T.M.G., cependant, pour des raisons indépendantes indépendantes de notre volonté, volonté, l’accès à cet événement nous fut refusé. De même, nous ne pûmes assister à aucun C.D.C. C.D.C. ("Conseil ("Conseil De Classe"). Classe"). Tout ceci nous permit de faire de multiples observations instructives, selon les axes décrits ci-après. 2.1
Le choi choix x des des axes axes d’ob d’obse serv rvati ation on
Nous avons, conformément aux instructions en vigueur, effectué chacun une observation ciblée selon deux axes préétablis à l’avance. Nous avions préparé des fiches préalablement remplies pour être plus efficaces dans notre observation au
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Stage de Pratique Accompagnée
CRCG
parfois moins, sous préférons faire une synthèse des résultats. Pourquoi Pourquoi avons avons nous choisi ces axes d’observation d’observation ? • La discipline : J’ai choisi cet axe d’observation car la pratique d’une bonne discipline est un sujet qui me tient particulièrement à cœur. Même si les professeurs se plaisent à dire qu’ils ne sont pas là pour faire la police, cela fait en pratique pleinement partie de leurs charges, souvent contre leur gré. Ainsi, une bonne pratique de la discipline permet une classe tenue, et une classe tenue permet un enseignement enseignement soutenu. J’ai donc un intérêt profond pour comprendre les mécanismes derrière cette science : les stratégies qui fonctionnent, ainsi que celles qui ne fonctionnent pas. Cet axe est celui qui est le point de départ à une classe attentive et intéressée, tous les autres axes découlent donc de celui-ci. • Les différents types d’activité : J’ai choisi cet axe d’observation car je pense que l’alternance entre différents types d’activités est une clef permettant de mieux prendre ne compte la diversité des élèves et ainsi de mieux attirer leur attention. J’ai également choisi d’étudier l’activité réelle des élèves en fonctions des différentes activités car c’est un sujet qui me tient particulièrement à cœur. En effet, c’est une des vocations du professeur de réussir à intéresser ses élèves en tout temps, tout lieu. Il lui incombe de réussir à faire participer tout le monde, même les élèves les plus en difficulté. Ainsi, une plus grande implication de la part des élèves élèves permet p ermet un enseignement enseignement plus soutenu soutenu et mieux accueilli. J’ai donc un intérêt profond pour comprendre les mécanismes derrière la didactique des savoirs savoirs disciplinaires disciplinaires au lycée : les stratégies stratégies qui fonctionnent, fonctionnent, ainsi que celles qui ne fonctionnent pas. J’ai notamment lu avec attention le livre Parler pour que les élèves écoutent, écouter pour que les élèves parlent , de Adele Faber et Elaine
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Stage de Pratique Accompagnée
CRCG
moments d’inattention d’inattention dans le référentiel référentiel du cours ? • Quand se sont produits les moments Débu Débutt du cour courss Mili Milieu eu du cours cours Fin Fin du cours cours éléments déclencheurs déclencheurs de ces moments d’inattent d’inattention ion ? • Quels ont été les éléments 1. 2. 3. 4.
5. 6. 7. 8.
• Le professeur applique les punitions suivantes : ◦ Séparation des élèves ◦ Exercices supplémentaires ◦ Interrogation supplémentaire ◦ Correspondance avec les parents ◦ Exclusion ◦ Détention ◦ Autre moyens ont été utilisés par le professeur pour récupérer l’attenti l’attention on ? • Quels moyens 1.
5.
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Stage de Pratique Accompagnée
* + % . )
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Stage de Pratique Accompagnée
CRCG
journée peut être résumée dans le graphique suivant. suivant.
Figure 1.4
– Attention de l’élève en fonction de l’heure de la journée.
Ainsi, le matin est plus propice à un cours magistral et l’après-midi à un travail dirigé. C’est donc ce que nous avons essayé de faire en pratique une fois la phase d’observation terminée. • Ce qui est ressorti du deuxième point d’observation est assez étrange. En effet, les moments d’inattention se sont plus fréquemment produits dans l’ordre : en milieu de cours, en début de cours et enfin en fin de cours. A quoi cela est-il dû ? Il est possibl p ossible, e, voire voire probable, que la plage du "milieu" de cours cours soit soit temporel temporellem lemen entt plus plus large large que celle du début début ; c’est c’est ainsi ainsi fort fort naturel naturel que le plus grand nombre de moments d’inattention se concentre ici . En proportionnalité moment d’inattention sur temps de la plage, nous ne sommes pas
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Stage de Pratique Accompagnée
CRCG
◦ Correction des exercices événement sortant sortant de l’ordinaire l’ordinaire arrive arrive ◦ Dès qu’un petit événement ◦ Effacement trop rapide du tableau Nous pouvons ainsi en conclure que chaque petite chose est propice à l’inattention. • Listons les différentes punitions qu’applique le professeur, toujours avec leur fréquence d’apparition.
◦ Correspondance avec les parents ◦ Exclusion (menace uniquement, si carnet non signé ou exercices non faits) Nous remarquons que le nombre de punition est donc très faible. Les élèves doivent être très sages? • Listons les différents moyens qu’emploie le professeur pour récupérer l’attention de ses élèves, encore avec leur fréquence d’apparition.
◦ Demande explicite de silence ◦ Tape dans ces mains (plutôt fréquent) ◦ Chuuuuut (vraiment très fréquent) ◦ Réprimande nominatives (très fréquent) ◦ Ironie, sarcasme (super fréquent) ◦ Voix qui monte ◦ Faire une blague • Enfin, pour conclure ce beau travail, croisons le type de réaction du professeur
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Stage de Pratique Accompagnée 2.3
CRCG
Axe Axe : les différe différen nts types types d’ac d’activ tivité itéss
............................ Date : ............................ Activité
Recherche d’exercices Correction d’exercices Intervention Intervention d’élève Manipulation Debriefing contôle Interrogation
............................ Heure : ............................
............................ Classe : ............................
Temps Nombre Comm Commen entt les les élèv élèves es Oral/Écrit d’élèves participent participent-ils -ils ? participant
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Stage de Pratique Accompagnée
CRCG
Voici mes fiches d’observation (résumées).
TS Spécialité Date 10/10
Horaire 13h30 - 15h20
Classe Effectif TS spécia- 19 élèv élèves es lité
Enseignant Mme Mme Lefe Lefebbvre
Les différentes activités chronologiquement : ◦ Debriefing contrôle (10 min) : La correction est polycopiée, les élèves posent des questions questions à l’oral. l’oral. ◦ Correction des exercices (35 min) : Les élèves passent au tableau et répondent aux questions. Ils donnent donnent d’autres pistes à l’oral. l’oral. L’enseignan L’enseignante te passe au tableau tableau également. ◦ Recherche d’exercices (10 min) : Tous les élèves participent à l’écrit. ◦ Pause (2 min)
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Stage de Pratique Accompagnée
CRCG
◦ Debriefing devoir maison (10 min) ◦ Correction d’exercices (15 min) : Les élèves se lèvent, prennent le stylo et corrigent
au tableau. ◦ Cours (30 min) : Les élèves travaillent à l’oral sur des exemples simples puis généraux. Ils traitent beaucoup d’exemples. Ils découpent et collent un cours polycopié mais ils écrivent écrivent eux-mêmes les exemples. exemples. ◦ Pause (15 min) ◦ Recherche d’exercices (15 min) : Les élèves recherchent des exercices type. Ils les traitent à l’écrit, dans le cours. ◦ Cours (15 min) ◦ Recherche d’exercices (5 min) ◦ Cours (5 min) ◦ Recherche d’exercices (15 min)
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Stage de Pratique Accompagnée
CRCG
2nde -AP Date 21/11
Horaire 12h30 - 13h25
Classe 2nde AP
Effectif 11 élèves
Enseignant M. Vargas
Les différentes activités chronologiquement :
◦ Accueil, distribution des feuilles (5 min) ◦ TP algorithmique (10 min) : La feuille de cours/TD est polycopiée. Les élèves
travaillent au brouillon. L’enseignant passe dans les rangs et corrige au tableau. Les élèves répondent et posent des questions. ◦ Rappels de cours (5 min) ◦ TP algorithmique (10 min) ◦ Correction de l’exercice (15 min) implémentent l’algorithme sur leur calculatrice, ◦ Manipulation (10 min) : Les élèves implémentent
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Stage de Pratique Accompagnée
3
CRCG
Phas Ph ase e de prat pratiq ique ue acc accom ompa pagn gnée ée L’adaptabilité s’acquiert par l’expérience du changement. Daniel Jouve
3.1 3.1
Un cou cours rs de Eli Elio o
• Titre de la séquence en cours : Les nombres complexes. • Déroulement de la séquence et place de la séance : La séquence se déroule
en quatre séances de cours consacrées comme ceci : algèbre, équations, géométrie, trigonométri trigonométriee ; suivies d’une séance de devoir sur table, puis de quelques quelques séances d’annales; le tout agrémenté d’un devoir maison. La séance étudiée / analysée plus précisément ici est : la première séance de cours, autrement dit celle portant sur l’algèbre. • Objectifs de la séance : Les objectifs de la séance sont les suivants : développer leurs premières premières connaissances, connaissances, essentielles essentielles pour la ◦ Aider les élèves à développer
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Stage de Pratique Accompagnée
CRCG
• Difficultés prévisibles pour les élèves et stratégies envisagées : De nombreux pièges très classiques furent repérés dans la préparation de ce cours. Un √ premier est l’envie inhérente aux élèves d’écrire i = −1 sachant que i = −1. 2
Nous insisterons lourdement sur ce point. Un autre est le fait que la partie réelle et la partie imaginaire sont des nombres réels. Nous insisterons lourdement sur ce point. Si les élèves ont retenu cela en sortant du cours, nous nous considérerons comme victorieux. Enfin, une difficulté prévue est la capacité d’acceptation des élèves à l’introduction d’un nouveau nombre i . Qu’est-ce donc que ce nombre qui
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Stage de Pratique Accompagnée
CRCG
révèle partie d’un Tout appelé Globalité. Cette séance est le berceau mental qui servira de terre arable à la construction à venir dans l’esprit de nos élèves. Nous sommes questionnés sur la meilleure façon d’introduire le corps des complexes. Le bulletin officiel sus-mentionné préconisant l’introduction dans ce chapitre d’éléments lui donnant une dimension historique, c’est cette approche que nous avons choisie. En effet, en terminale scientifique, les nombres complexes sont censés être vus essentiellement comme constituant un nouvel ensemble de nombres avec ses opérations propres. Cette introduction s’inscrit dans la perspective d’un
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Stage de Pratique Accompagnée
3.2 3.2
CRCG
Un cour courss de de Mar Margu guer erit ite e
• Titre de la séquence en cours : Les nombres complexes. • Déroulement de la séquence et place de la séance : La séquence se déroule
en quatre séances de cours consacrées comme ceci : algèbre, équations, géométrie, trigonométri trigonométriee ; suivies d’une séance de devoir sur table, puis de quelques quelques séances
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Stage de Pratique Accompagnée
CRCG
Nous insisterons également sur le calcul du module d’un nombre complexe à partir de sa forme algébrique afin d’éviter les écueils "classiques" : résultats complexes, simplification simplificationss abusives abusives des racines,. . .. • Déroulement prévu (les différentes étapes et leurs durées ( sic), éventuellement une organisation particulière de la classe, du tableau, supports, TICE..... (sic )) : Les différentes étapes prévues sont les suivantes.
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Stage de Pratique Accompagnée
CRCG
Après y avoir réfléchi, nous choisissons de nous laisser parler de "plan complexe" pour désigner le "plan d’Argan-Cauchy", quand bien même le plan complexe désigne rigoureusemen rigoureusementt C2 , R-espace vectoriel de dimension 4. Comme les élèves ont peu de risque de faire la confusion (la notion d’espace vectoriel ne figure plus au programme du bulletin officiel bi-sus-mentionné depuis de bien (trop) nombreuses années), nous ne pensons pas que cette distinction soit appropriée dans notre contexte. Afin d’obtenir une activité réelle optimale des élèves pendant ce travail complexe,
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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Nombres Complexes
À rendre le lundi 21 novembre 2016
1. Résoudre dans C l’équation Z 2 + 4Z + + 16 = 0. C’est une équation polynomiale de degré 2 , on calcule ∆ = b
2
On a en utilisant les propriétés du cours sur les modules : z z¯ 4 + 4¯ z z 2 + 16 = z 4 + 4z 2 + 16
− 4ac = 16 − 4 × 1 × 16 = −48.
= z 4 + 4z 2 + 16 = z 4 + 4z 2 + 16 = ¯0 = 0
D’après le cours, les solutions sont donc
√ √ √ √ √ − b − i −∆ −4 − i 48 −4 − 4i 3 − 4 4 3 = = = − i = −2 − 2i 3 Z 1 = 2a
2
2
2
2
√
et Z 2 = Z 1 = −2 + 2i 3.
Remarque : Quand on sait que Z 1 = Z 2 , il est inutile de calculer séparément Z 1 et Z Z 2 !
√
2. On pose a = 1 + i 3. (a) Calculer a 2 . On a
√ √
a2 = (1 + i 3)2
√
= 1 + 2 i 3 + (i 3)2
√ = 1 − 3 + 2i 3 √ = − 2 + 2 i 3. (b) En déduire les solutions dans z 2 =
C de
l’équation :
√
−2 + 2i
3.
On a
√ −2 + 2i 3 ⇐⇒ z 2 = a 2 ⇐⇒ z 2 − a2 = 0 + a) = 0 ⇐⇒ (z − a)(z + = a ou z = −a. ⇐⇒ z = Donc l’ensemble des solutions de cette équation est S = {−a,a}. z 2 =
3. Démontrer que si z est une solution de l’équation ( E ), alors son conjugué z est également solution de (E ). En déduire les solutions dans C de l’équation (E ). On admettra que ( E ) admet au plus quatre solutions.
Supposons que z soit une solution de ( E ), montrons que ¯z est aussi une solution z est de ( E ).
car z est solution de ( E ) donc z 4 + 4z 2 + 16 = 0 . Donc ¯z est aussi solution de ( E ). z est Posons Z = z 2 , l’équation ( E ) se réécrit Z 2 + 4Z + + 16 = 0 . On a montré dans la question 1 qu’il y avait deux solutions à cette nouvelle équation : a 2 et a 2 = ¯a2 (d’après le calcul de a 2 effectué dans la question 2 (a)). On a donc Z = a 2 soit z 2 = a 2 , équation que nous avons résolue à la question 2 (b) avec pour solutions −a et a . Donc −a et a sont deux solutions de ( E ). D’après la première partie de la question, on a aussi que − a et ¯a sont deux solutions de a . On sait que : −a = −a¯. On vérifie facilement que − a,a, − a, ¯,a ¯ sont quatre solutions distinctes. Or ( E ) a admet au plus quatre solutions. Finalement, S (E ) =
{−a,a, − a,¯a,a¯} avec a = 1 + i√ 3.
Remarque Remarque culturelle : Une équation polynomiale de degré degré 4 qui ne contient pas de terme en z 3 et en z , c’est-à-dire de la forme az 4 + bz 2 + c = 0 est appelée appelée équation bicarrée. On peut toujours la résoudre en effectuant le changement de variable Z = z 2 . Autre Autre remarque remarque culturelle : Vous Vous savez résoudre des équations équations polynomiale de degré 1 ou 2 grâce aux formule apprises. Des formules du même genre bien que sensiblement plus compliquées existent pour des équations polynomiales de degré 3 ou 4 4 . Cependant, à partir du degré 5, il est impossible d’écrire de telles formules donnant les solutions dans le cas général.
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Nombres Complexes
Devoir en temps limité
Devoir en temps limité - Nombres complexes Exercice Exercice 1. Questions
diverses (6 points)
3 i . 6i 8 2. a) Trouver les racines de 8 z 2 + 12z + 5 .
− −
1. Mettre sous forme algébrique
b) En déduire les solutions dans 8(4iz
C de
− 3i + 3)
2
l’équation + 12(4iz
− 3i + 3) + 5 = 0. 3. Déterminer l’ensemble des points M tel que |z − 1 + i| = |z + 1 − i|. M
M
Exercice Exercice 2. Une
démonstration de l’inégalité triangulaire (6 points) Soit z et z deux nombres complexes. complexes. 1. Développer ( z + z )(z + z ), puis en déduire que
z + z 2 = |z |2 + z 2 + zz + z z.
2. a) Compléter z + z¯ = . .. quand z = a + bi. b) Calculer le conjugué de zz . c) En déduire que
2 2 z + z = |z |2 + z + 2 Re( Re(zz ).
d) On admet que Re( z ) |z |. En déduire que
2 z + z (|z | + z )2 .
e) En déduire l’inégalité triangulaire. Exercice Exercice 3. De
l’algèbre et de la géométrie (8 points) →u ,−→v ).√ Onconsidère les points A, 1. Le plan est muni d’un repère orthonormé direct (0 ,− d’affixes respectifs zA = 1 + i, zB = z A et zC = 2 2 cos( 74π ) + i sin( 74π ) . B et C d’affixes a) Déterminer la forme algébrique de zB et zC . b) Écrire sous forme trigonométrique zA et zB en justifiant. 2. Soit I le point d’affixe zI = 3. a) Déterminer l’affixe du point D tel que le quadrilatère AICD soit un parallélogramme en justifiant. √ b) Montrer que les points A et C sont sont sur le cercle C de centre I et de rayon 5. c) Déterminer la nature du triangle AIC AI C en justifiant. En déduire la nature du quadrilatère AICD.
−−→
−→
3. On considère le point E qui qui vérifie OE = 2I C et et le point F d’affixe zF = i × zE . a) Déterminer l’affixe zE du point E d’affixe d’affixe zF du point F sous forme algébrique. b) Montrer que les droites ( AB ) et ( C F ) sont perpendiculaires.
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Nombres Complexes
TS4
Devoir Surveillé Corrigé - Nombres complexes 3 6i
On a 3 6i
−i . −8
2
+ z | |z +
1
3 4
2
1 4
2
b) En déduire les solutions dans C de l’équation
2
|z + + z |
2
1
1
c) En déduire que
|| | |
2
= z 2 + z + zz + zz .
|| | |
Et d’après la question 2. a), zz + zz = 2Re(zz 2Re(zz ). 2 2 2 Ainsi, z + + z = z + z + 2Re(zz 2Re(zz ). d) On admet que Re(z Re( z ) z . En déduire que Montrons tout de même que Re(z Re(z ) z . On écrit z = a + bi + bi sous forme algébrique, ainsi z = a2 + b2 a2 = a a = Re(z Re(z ).
|
|
|| | | || | | √
√ | | || + z | (|z | + |z |) . |z + 2
8(4iz 8(4iz
− 3i + 3)
2
+ 12(4iz 12(4iz
− 3i + 3) + 5 = 0.0 .
L’équation L’équation est de la forme 8 forme 8Z Z 2 + 12Z 12Z + + 5 = 0 avec Z avec Z = = 4iz 3i + 3, 3, d’après 1 la question précédente, on a Z = Z = z 1 ou Z ou Z = z 2 , ssi 4 ssi 4iz iz 3i + 3 = 34 i 4 3 1 15 11 15 13 ou 4iz 3i + 3 = 4 + 4 i, ssi 4iz = + 4 i ou 4iz = + 4 i, i , ssi 4 4 15 15 z = = 16 + 11 ou z ou z = = 16 + 13 , ssi z ssi z = = 11 + 15 i ou z = z = 13 + 15 i. 16 16 16 16 16 16 i i 15 13 15 Finalement, S = 11 + i, + i . 16 16 16 16
−
− −
−
{
−
−
}
−
− − − −
3. Déterminer l’ensemble des points M tel que z M 1 + i = z M i. M M + 1 Soit A Soit A le le point d’affixe z d’affixe z A = 1 i et B et B d’affixe d’affixe z z B = 1+i 1+ i. On cherche l’ensemble des points M M vérifiant z M 1 + i = z M i soit z M z A = z M z B , M M + 1 M M soit AM soit AM = = BM , [ AB]]. BM , soit la médiatrice du segment [AB
− | − | |
| − | | −| − − | | − | | − |
Exercice 2. Une démonstration de l’inégalité triangulaire (6 points) Soient z et z et z deux nombres complexes.
2
= z 2 + z + 2Re(zz 2Re(zz ).
On applique la question 2. b) à l’égalité trouvée à la question 1, ce qui nous donne
− 4ac = ac = 12 − 4 × 8 × 5 = −16 16 < < 0 0.. √ −b − i −∆ = −12 − 4i = − 3 − 1 i. Première solution : z : z = 2a 16 4 4 Seconde solution : z : z = z = − + i. Finalement, S = {z , z }. 2
|| | |
2
2. a) Trouver les racines de 8z 8 z 2 + 12z 12z + + 5 . Calcule de discriminant : ∆ = b2
2
= z 2 + z + zz + z z.
On a z z = zz = zz = z z .
18ii + 8i 8i + 6i 6i 10ii 3 1 − i = (3 − i)(−8 − 6i) = −24 − 18 −30 − 10 = = − − i. 6 + (−8) 6 + (−8) 100 10 10 −8 2
2
||
2
2
+ z | |z +
On a (z ( z + + z )(z )(z + + z ) = (z + + z )(z )(z + + z ) = zz + + zz + z z + + z z . D’après D’après le cours, cours, 2 pour tout nombre complexe z complexe z , zz = zz = z , ce qui nous donne bien l’égalité voulue. 2. a) Compléter z + z + ¯z z = = . . . quand z = z = a + bi. D’après le cours, z cours, z + + z = = 2a. b) Calculer le conjugué de z z .
Exercice 1. Questions diverses (6 diverses (6 points) 1. Mettre sous forme algébrique
1. Développer (z ( z + + z )(z )(z + + z ), puis en déduire que
2
On reprend l’égalité de la question 2. c), ce qui nous donne car Re(zz Re(zz ) zz :
2
2
2
2
2
|z + + z | = |z | + |z | + 2Re(zz 2Re(zz ) |z | + |z | + 2 zz . Or zz = | z | z , une identité remarquable donne alors que | z | + |z | + 2 |z | z = ( |z | + |z |) , ainsi |z + + z | (|z | + |z |) . e) En déduire l’inégalité triangulaire. La fonction fonction x → croissantee sur R , ainsi on peut l’appliquer sur √ x est croissant
2
2
2
2
2
+
l’inégalité de la question 2. d) sans changer le sens de l’inégalité. On a donc
|
que pour tous nombres complexes z et z , z + + z z + + z et z + z sont deux réels positifs, donc
|
| | | | | + z | |z | + |z | |z +
|
2
| |
(Inégalité triangulaire) triangulaire).
( z + z )2 . Or
| |
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Nombres Complexes
TS4
grâce à l’argument de Remarque : On aurait aussi pu calculer l’angle AIC AIC z −z . z −z De plus, d’après la question précédente, AI = AI = CI . CI . Donc le triangle AI C est rectangle isocèle en I . I . Un parallélogramme qui a un angle droit et deux côtés consécutifs de même longueur est un carré. Donc le quadrilatère AICD est AICD est un carré. C
I
A
I
−−→
−→
3. On considère considère le point E point E qui qui vérifie OE = = 2IC et point F d’affixe d’affixe z IC et le point F z F F = i
× z . E E
l’affixe z F a) Déterminer l’affixe z E E du point E et l’affixe F du point F sous forme algébrique.
−−→
−→ ⇐⇒ ⇐⇒ − − −
− − → = z − → − − → = 2z − → OE = 2IC z OE z OE z E z O = 2(z 2(z C E C IC 2IC z I I ) z E 2(z C z I I ) z E 2i 3) = 2( 1 2i) = 2 4i. E = 2(z C E = 2(2 Ainsi z Ainsi z E 2 4i. Donc z Donc z F z E i ( 2 4i) = 2i 4i2 = 4 2i. E = F = i E = i( b) Montrer que les droites (AB ( AB)) et (CF ( CF )) sont perpendiculaires. z F z C F C On calcule calcule l’angle l’angle orienté orienté (AB, CF ) CF ) = arg modulo 2π. Or z B z A z F z C 4 2i 2 + 2i 2i 2 F C = = = i. Or Or arg(i arg(i) = π/2 π/2 modulo 2π 2 π, donc z B z A 1 i 1 i 2i l’angle ( l’angle ( AB, CF ) CF ) est un angle droit. Finalement, (AB ( AB)) (CF ) CF ). Remarque : On aurait aussi pu calculer un produit scalaire.
⇐⇒ ×
− − − −
−−→ −−→
⇐⇒
− − − − −−→ −−→− − −
⇐⇒ − − − − −
−
− −
−
− − −
⊥
Question Bonus : A faire si, et seulement si, vous avez traité le reste du sujet.
Résoudre l’équation z 3 + 5z 5z 2 + 4z 4z = = 10. On a z 3 + 5z 5z 2 + 4z 4z = = 10 z 3 + 5z 5z 2 + 4z 4z 10 = 0 P ( P (z ) = 0. On remarque que 1 que 1 est solution de cette équation. On cherche donc P de P de la forme :
⇐⇒
−
⇐⇒
2
P ( P (z ) = (z
− 1)(az 1)(az + bz + + c) On développe : ( : (z z − 1)(az 1)(az + bz + + c) = az + z (b − a) + z (c − b) − c. Donc par unicité des coefficients de P , P , on a a a a = 1, 1 , b = b = 6 et c = c = 10. 10 . Ainsi, z Ainsi, z +5z +5z +4z +4z = = 10 ⇐⇒ (z −1)(z 1)(z +6z +6z +10) +10) = 0 ⇐⇒ z −1 = 0 ou z +6z +6z +10 +10 = 2
3
3
2
2
2
2
0. On résout l’équation du second degré z 2 + 6z 6z + + 10 = 0 dans 0 dans C . ∆ = b2 4ac 4 ac = 36 40 = 4 < 0. Ainsi cette équation admet deux solutions complexes conjuguées : b i ∆ 6 2i z 1 = = = 3 i et z 2 = z = z 1 = 3 + i. 2a 2 Finalement, l’ensemble solution de l’équation z 3 + 5z 5z 2 + 4z 4z = = 10 10 est est :
− − − √ −
− − − − − − S =
−
{1, − 3 − i,3 i,3 + i}.
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Nom et Prénom : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interrogation de Cours a) Si z est un nombre complexe de point image M, que représente géométriquement | z | ? ............................................................................................................. b) Soit Soit
z = 3 + i .
Calculer Calculer
zz ¯.
..................................................................................
............................................................................................................. c) Soient Soient A et B deux points points du plan d’affixes respectifs respectifs Calculer
z−→ = AB
zA = 2 − 3 i et zB = 17i + 4.
.............................................................................................
d) Calculer Calculer (sans mettre sous forme algébrique) algébrique) i − 4 =................................................................................................... 3 − 4 i
e) Quel est l’ensem l’ensemble ble des points de M d’affixe z tels que z − z
= 0?
.............................................................................................................
Nom et Prénom : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interrogation de Cours a) Si z est un nombre complexe de point image M, que représente géométriquement | z | ? ............................................................................................................. b) Soit Soit
= 1 + 4i
Calcul
¯
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TS4
Résolution d’équations dans C Exercice Exercice 1.
Résoudre les équations suivantes dans C et écrire les solutions sous forme algébrique : ¯ = 1 − 3z ¯ . 1. (1 + i )z z z ¯ + + iz = 1. 2. z z ¯ + + z = 2. 3. z z Exercice Exercice 2.
a) Sans utiliser ∆, résoudre les équations suivantes dans C : 0. 1. z 2 + 1 = 0. 2. 16z 2 = −3. b) Résoudre les équations suivantes dans
C
:
+ 13 = 0. 0. 1. z 2 + 6z + 0. 2. z 2 + 3z − 10 = 0. 0. 3. −z 2 + 8z − 25 = 0. Exercice Exercice 3.
On cherche à résoudre l’équation (E ) d’inconnue complexe z : 3
z
− z 2 + z − 1 = 0.
¯ est aussi une solution. 1. Démon Démontre trerr que si z est une solution de (E ), alors z z 2. Vérifie Vérifierr que 1 que 1 et e t i sont deux solutions de (E ). 3. En admettant admettant qu’une équation polynomiale polynomiale complexe de degré n a au plus n solutions, déterminer l’ensemble des solutions de (E ). Exercice Exercice 4.
+ 4i). 1. Développer Développer (iz − 2)(z + 2. En déduire déduire les racine raciness de iz 2
6
8i ( 2i et
4i)
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TS4
Résolution d’équations dans C Exercices
Exercice Exercice 1.
Résoudre les équations suivantes suivantes dans C et écrire les solutions sous forme algébrique :
4 + 7i = ⇐⇒ z = 1 + 2i ⇐⇒ z = (4 + 71i+)(14 + 2i) = (4 + 8i +5 7i − 14) = −2 + 3i. −1 + 2i = (1 + i)z + + 1 ⇐⇒ z (−2 + i) = −1 + 2 i ⇐⇒ 2. 2i + 3 z = −2 + i ⇐⇒ z = (−1 +−22i)(+− i2 + i) = (2 − 2i −5 4i − 2) = − 65 i. (1 + 2i) 1. z (1
− 1 = 7i + 3 ⇐⇒
z¯ = 1 3. (1 + i)z
− 3z z¯ ⇐⇒
(4 + i) = 1 z z¯ (4
z z¯ =
(4 i) 4 = 16 + 1 17
⇐⇒ + iz = 1 z¯ + 4. z
(1 + 2i) = 4 + 7i z (1
⇐⇒
= z z¯ =
1 4 + i
− 171 i ⇐⇒
−
= z =
4 1 + i. 17 17
2
⇐⇒ a − bi + ai + bi = 1 ⇐⇒ (a − b) + (a − b)i = 1 ⇐⇒ a − b = 1 et a − b = 0
Il n’y a pas de solution dans C. z¯ + a = 1. + z = 2 2a = 2 5. z Il y a une infinité de solutions dans C : ce sont les complexes de la forme 1 + bi avec b un réel quelconque.
⇐⇒
Exercice Exercice 2.
⇐⇒
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TS4
− 10 = 0, ∆ = 9 + 40 = 49, z = −32± 7 . −6 ± 4i . + 13 = 0, ∆ = 36 − 52 = −16, z = 3. z + 6z + 2 −8 ± 6i . 4. −z + 8 z − 25 = 0, ∆ = 64 − 100 = −36, z = −2 2. z 2 + 3z 2
2
Exercice Exercice 4.
On considère le polynôme :
+ 4 i. P (z ) = 7z 5 + (5 + 7i)z 4 + (29 + 5i)z 3 + (20 + 29i)z 2 + (4 + 20i)z + 1. Vérifier Vérifier que que z =
−i est solution de l’équation P (z ) = 0. Trivial.
2. Soit Soit z dans C, alors :
(z + + i)(z 2 + 4)(7z 2 + 5z + + 1) = (z 3 + 4z + + iz 2 + 4i)(7z 2 + 5z + + 1) = 7z 5 + 5z 4 + z 3 + 28z 3 + 20z 2 + 4z + + 7 iz 4 + 5iz 3 + iz 2 + 28iz 2 + 20iz + + 4i = 7z 5 + (5 + 7i)z 4 + (29 + 5i)z 3 + (20 + 29i)z 2 + (4 + 20i)z + + 4 i = P (z ) 3. En déduire déduire toutes toutes les solution solutionss dans C de l’équation P (z ) = 0. Les solutions de l’équation P (z ) = 0 sont z = i et les solutions des équations :
−
+ 1 = 0. z 2 + 4 = 0 et 7z 2 + 5z + (z 2i)(z + + 2i) = 0. Or on a : z 2 + 4 = 0 De plus, on considère l’équation :
⇐⇒ −
7z 2 + 5z + + 1 = 0 ∆ = 25
− 4 × 7 × 1 = 25 − 28 = −3
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TS4
Nombres complexes et géométrie Exercice Exercice 1.
On considèr considèree quatre quatre points points A,B,C,D d’affixes respectifs respectifs z A = −2, z B = 1 + i, z C − 1 − 2i et z D = 2 − i. C = −1 1. Faire un dessin. dessin. Remar emarque : De manièr manièree génér générale ale,, que ce soit soit demand demandéé ou non, non, il est toujours toujours extrêmement recommandé de faire un dessin lorsqu’on a à résoudre un problème de géométrie. 2. Montrer Montrer que que ABCD est un parallélogramme. 3. Que le triang triangle le ODC est rectangle isocèle. 4. Détermin Déterminer er l’affixe de H centre de ABCD .
5. Détermin Déterminer er l’affixe de C symétrique de C par rapport à D . Exercice Exercice 2.
Déterminer l’ensemble des points M d’affixe z du plan tels que 1. |z | = 4, 2. arg(z ) = − 3.
π
4
z + 1 ∈ iR. z − 1
[2π ],
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TS4
Nombres complexes et géométrie
Exercice Exercice 1.
On considère quatre points A,B,C,D d’affixes A,B,C,D d’affixes respectifs z A = −2, z B = 1 + i, z C = −1 − 2i et z et z D = 2 − i. 1. Faire un dessin. dessin. Remarque : De manière générale, que ce soit demandé ou non, il est toujours extrêmement recommandé de faire un dessin lorsqu’on a à résoudre un problème de géométrie.
2. Montrer Montrer que que ABCD est ABCD est un parallélogramme. −→ = z B − zA = 1 + i − (−2) = 3 + i, et z −−→ = z D − zC = 2 − i + 1 + 2i On a z AB et z CD 2i = 3 + i, donc les affixes sont égaux, donc les vecteurs sont égaux, donc on a un parallélogramme. 3. Que le triang triangle le O ODC DC est est rectangle isocèle. (2−i)(− )(−1+2i) zD −zO On calcule z calcule z = zC −zO = −21−−i2i = (2− = 5 Avec les propriétés du module, |z | z | =
zD −zO z −z C
O
=
5i i . 5 = i. |zD −zO | |zC −zO |
=
OD OC
.
Or | Or |zz | = |i| = 1, donc O donc OD D = O = OC C , donc le triangle est isocèle en O. O . −−→ −−→ De plus on a OC d’affixe OC d’affixe 2 − i et OD d’affixe −1 − 2i, donc on a les coordonnées. On −−→ −−→ calcule le p.s. OC .OD = OD = −2 + 2 = 0. 0. 4. Détermine Déterminerr l’affixe de H de H centre centre de ABCD. ABCD . H H est le milieu des diagonales, donc H H milieu de [AC ], donc zH = 1 3 2 (−2 − 1 − 2i) = − 2 − i.
1 2 (zA
+ z C ) =
5. Détermine Déterminerr l’affixe de C de C symétrique de C par C par rapport à D. D . 1 ), soit 2(2 On a D milieu de [C [ C C ], donc z donc z C vérifie z vérifie z D = 2 (zC + zC soit 2(2 − i) = −1 − 2i + zC
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Nombres complexes et forme trigonométrique Exercice 1.
Déterminer l’ensemble des nombres complexes 1.
| | = 4,
:
z
2. arg(z ) = 3.
z vérifiant
z + + z
1 1
− ∈
π
−4
[2π],
iR.
Exercice 2.
Mettre sous forme trigonométrique les complexes suivants : 1 + i , 3 Mettre sous forme algébrique les complexes suivants : z =
√ 2(cos( 5
π
6
) + i sin(
z =
5π )). 6
π
−4
i
et i. π
Exercice 3.
1. Mettre Mettre sous forme trigonométri trigonométrique que (1 + i )8 . 2. Détermin Déterminer er tous les entiers entiers
n tel
π
π
3(cos( ) + i sin( ), z = cos( ) + i sin( ) et 3 3 2 2
que (1 + i ) est réel. n
Nombres complexes et forme trigonométrique
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TS4
Sujet Bac Antilles-Guyanne 2013 On considère la suite (z ) à termes complexes définie par entier naturel n, par
z 0
n
z n+1
=
= 1 + i + i et, pour tout
z n + z n
3
| |.
+ ib , où a est la partie réelle de Pour tout entier naturel n, on pose : z = a + i et b est la partie imaginaire de z . Le but de cet exercice est d’étudier la convergence des suites (a ) et (b ). n
n
n
n
n
n
n
Partie Partie A
1. Donner
a0
et
b0 .
2. Calcul Calculer er z 1, puis en déduire que
a1
√
1+ 2 = et 3
b1
1 = . 3
3. On considère l’algorithme l’algorithme suivant suivant : Variables :
A et e t B des
nombres réels K et N des nombres entiers Initialisation : Affecter à A la valeur 1 Affecter à B la valeur 1 Traitement : Entrer la valeur de N Pour K variant de 1 à N A + A2 + B 2 Affecter à A la valeur
√
n
z n
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TS4 ( b ) ? En déduire l’expression de b en fonction 2. Quelle est est la nature nature de la suite suite ( de n, et déterminer la limite de (b ). n
n
n
3. (a) On rappelle que pour p our tous nombres nombres complexes z et
+ z | |z | + |z | |z +
z
:
(inégalité triangulaire).
Montrer que pour tout entier naturel n,
|z | 2 |3z | . n
+1
n
(b) Pour Pour tout entier naturel naturel n, on pose u = z . Montrer par récurrence que, pour tout entier naturel n,
| |
n
n
2 √ n
un
3
2.
En déduire que la suite ( suite (u ) converge vers une limite que l’on déterminera. n
(c) Montre Montrerr que, que, pour tout entier entier nature naturell n, a u . En déduire que la suite (a ) converge vers une limite que l’on déterminera.
| | n
n
n
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Un optimiste, c’est un pessimiste qui a beaucoup d’expérience. Possiblement Olivier Fouquet, Quelque part ailleurs.
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